# Questions tagged [mgf]

The moment generating function (mgf) is a real function which allows to derive a random variable's moments and therefore can characterize its entire distribution. Use also for its logarithm, the cumulant generating function.

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### Why is moment generating function (MGF) evaluated at zero?

Although this question touched on MGF exists at neighborhood of 0, I still don't understand why the definition of MGF says $M_X{(t)}$ = E $e^{tX}$ for $t$ in neighborhood of 0. Why must it be 0 but ...
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### Understanding t bounds on MGF

I'm having trouble understanding what the bounds of the $t$ variable are for an mgf. My questions are bolded. Here's an example from a textbook: Suppose X is a random variable for which the pdf is: ...
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### By conditioning on $N$, show that the moment generating function of $Y$ is given by $m_Y(t)=m_N(\ln(m_X(t)))$

I am having a difficult time using moment generating function properties to prove this: (any direction or key properties will be very helpful) Let $X_1$, $X_2$, . . . be independent and identically ...
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### Saddlepoint approximation with weibull distribution

I have some trouble with this computation, I have the moment generating function of a random variable $S$ by: $$M_S(t)=\frac{\beta\mu t}{1+(1+\beta)\mu t-M_X(t)}$$ According to the text that I am ...
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### Skewness of Tweedie distribution

Tweedie distributions are a family of distributions from the exponential dispersion family that have power-law mean-variance relationship: \begin{align} \mathbb E[X] &= \mu \\ \operatorname{Var}[...
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### How to find the support of MGF

I am having a difficult time determining the support or range of a MGF for a given pmf. The specific questions states to find the MGF of f(x)=6/((x^2)(pi^2)) for x=1,2,3... The result (which is the ...
The current Wikipedia definition is The distribution of a random variable $X$ with distribution function $F$ is said to have a heavy (right) tail if the moment generating function of $F,$ $MF(t),$ ...
If $Y= \log10{(X)}$ .Y follows $N(\mu,\sigma^2)$ and has the mgf \$My(t)=e^(5*t+2*t^2) Then P(X<1000)=? The far I got is that after comparing the given mgf with that of normal we get mu=5 and sigma^...